(From a Copenhagen interpretation perspective) it is not the system that "jumps" but rather the fact that we must observe the system to say anything about it and we of necessity will have gaps between observations, especially when measuring different (complementary) observables. Understanding that the wave functions we write down are representations of our knowledge about the potential future observations based on the last actual observation made (or asserted) we, upon actually making an observation see in parallel a sudden update in our information and a sudden update in our description.

Think of a movie played in slower and slower speeds, eventually the smooth motion you conceive of resolves as a discrete sequence of pictures in which the players positions jump form one frame to the next. Now add to this the quantum business of only knowing what we have observed we can't really say the players moved smoothly from one position to the next between frames, (nor can we say they did not!!!) All we can say is that at the points of observation there are distinctions. Things change and so when looking at a discrete sequence of observations we see jumps. Then when we try to model what we see we find we cannot create a smoothly changing model of "what is" that correctly predicts the probabilities of outcomes for all potential choices for our sequence of what to observe.

This ties into the fact that to physically know something about a system you must physically interact with it and all interactions are of necessity two-way. We cannot be affected by a system (thereby getting knowledge from it) without affecting the system (thereby invalidating other knowables). So to try to resolve between the jumps we must continuously observe and in so doing we dramatically change what will happen. We are not talking about the same situation as before when we waited between acts of measurement.

There are physical implications to this, such as quantum tunneling. As we describe the energy of certain systems and the continuous transition between two stable states, there is not enough energy for the system to be observed in an intermediate state even though it has plenty for the two stable states. If we in fact continuously measured the energy of the system it would indeed be impossible for it to make the transition between the two states. But if we choose not to observe the energy (noting that this is a physical choice to physically treat the system differently) we will see a certain non-zero probability for the transition between stable states to occur. We cannot say what happen in the intermediate period because such a statement is directly tied to the assertion that we were interacting with the system via an act of observation. That is the meaning we give to saying something happened at that point. To understand the tunneling process we must shift to a distinct paradigm which invalidates our previous definition of energy.

[We can for example describe the potential energy as a statistical quantity expressing an aggregate expected value for say photon exchanges between system and environment. There is then a certain probability of "sneeking past the guards" without interacting with this (now statistically described) energy barrier. It is a complementary paradigm analogous to (and intimately connected with) the complementarity of some observables.

In this alternative paradigm the system never jumps but the dynamical environment is no longer continuously described. One "pulls on one thread an another pops up". One then realizes it isn't a matter of "picking the right thread" but taking a step back and understanding the futility of the attempt, rather looking at how it all knits together.]